The Medrash explains how each of the Begadim that were worn by the Kohen Gadol was a Kapara for a different Aveira. Two of them were the Michnasayim and the Tzitz. The Gemora says the Michnasayim were mechaper on the aveira of Arayos, and the Tzitz was mechaper on "Gasus Haruach" which means being haughty. The Kli Yakar says something interesting. He says that the tzitz was also a Kapara for Arayos. But the difference was that the michnasayim were a Kapara for Arayos in a hidden way, and the tzitz was for Arayos out in the open - brazenly - for this goes into the category of Gasus Haruach. This is also indicated by the fact that the Tzitz was worn out in the open, and the Michnasayim were worn hidden under the other garments.
How were these begadim a Kapara for different Aveiros?
I believe the answer is, when we show Hashem that we are truly interested in doing Teshuva, that is the greatest Kapara. One of the best ways to stop ourselves from doing an aveira that we have become accustomed to, is to set ourselves reminders in obvious places to prevent us from slipping up. By wearing these Begadim in these locations, the Kohel Gadol reminded us to control our YH in these areas, and that is a real step towards overcoming our Nisyonos.
The location of those two Begadim is very significant. There are 2 aspects to this addiction. The thought process - reading, seeing, thinking... - and the physical aspects... By wearing a Beged on the head, it is to remind us to control our thoughts, and the Beged on the Guf is to remind us to control our physical urges.
Everybody must set up their own reminders in their own places, in their own style, geared specifically towards their own YH. And we must know and remember that BY TAKING REAL STEPS TOWARDS FIXING THE PROBLEMS, WE ARE ALREADY EARNING A MEASURE OF KAPARA.
Just to end on a side note. When we wear our Tefillin - by our hearts and on our heads - it is also for the purpose of controlling our desires and our thoughts. It would not be a bad idea to use the time when we are wearing our Tefillin to ask Hashem for some extra help in our struggles.